Formula:3926

Digital Library of Mathematical Functions ID 10.19.E9

${\displaystyle{\displaystyle\rselection{{H^{(1)}_{\nu}}\left(\nu+a\nu^{\frac{1}{3}}\right)\\ {H^{(2)}_{\nu}}\left(\nu+a\nu^{\frac{1}{3}}\right)}\sim\frac{2^{\frac{4}{3}}}{\nu^{\frac{1}{3}}}e^{\mp\pi i/3}\mathrm{Ai}\left(e^{\mp\pi i/3}2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{P_{k}\left(a\right)}{\nu^{2k/3}}+\frac{2^{\frac{5}{3}}}{\nu}e^{\pm\pi i/3}\mathrm{Ai}'\left(e^{\mp\pi i/3}2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{Q_{k}\left(a\right)}{\nu^{2k/3}},}}$

Symbols List

• ${\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}$:
• ${\displaystyle{\displaystyle{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}$: Bessel function of the third kind (or Hankel function)
• ${\displaystyle{\displaystyle{H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}$: Bessel function of the third kind (or Hankel function)
• ${\displaystyle{\displaystyle\sim}}$:
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)}}$:
• ${\displaystyle{\displaystyle Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)}}$:
• ${\displaystyle{\displaystyle k}}$: nonnegative integer
• ${\displaystyle{\displaystyle\nu}}$: complex parameter
 

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